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A cyclic iterative method for solving multiple sets split feasibility problems in Banach spaces. (English) Zbl 1442.65111

Summary: In this paper, we construct an iterative scheme and prove strong convergence theorem of the sequence generated to an approximate solution to a multiple sets split feasibility problem in a \(p\)-uniformly convex and uniformly smooth real Banach space. Some numerical experiments are given to study the efficiency and implementation of our iteration method. Our result complements the results of F. Wang [Numer. Funct. Anal. Optim. 35, No. 1, 99–110 (2014; Zbl 1480.47102)], 99-110), F. Schöpfer et al. [Inverse Probl. 24, No. 5, Article ID 055008, 20 p. (2008; Zbl 1153.46308)], and many important recent results in this direction.

MSC:

65K10 Numerical optimization and variational techniques
49M37 Numerical methods based on nonlinear programming
90C25 Convex programming
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[1] Alber, Y. I., Metric and generalized projection operator in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, 178, 15-50 (1996), Lecture Notes in Pure and Applied Mathematics: Dekker, New York, Lecture Notes in Pure and Applied Mathematics · Zbl 0883.47083
[2] Byrne, C., Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18, 2, 441-453 (2002) · Zbl 0996.65048 · doi:10.1088/0266-5611/18/2/310
[3] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 1, 103-120 (2004) · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[4] Cegielski, A., Iterative Methods for Fixed Point Problems in Hilbert Spaces, 2057 (2012), Springer: Springer, Heidelberg · Zbl 1256.47043
[5] Censor Snf, Y.; Elfving, T., A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8, 2-4, 221-239 (1994) · Zbl 0828.65065 · doi:10.1007/BF02142692
[6] Censor, Y.; Lent, A., An iterative row-action method for interval convex programming, J. Optim. Theory Appl., 34, 321-353 (1981) · Zbl 0431.49042 · doi:10.1007/BF00934676
[7] Cioranescu, I., Geometry of Banach spaces, duality mappings and nonlinear problems (1990), Kluwer Academic: Kluwer Academic, Dordrecht · Zbl 0712.47043
[8] Dunford, N.; Schwartz, J. T., Linear Operators I (1958), WileyInterscience: WileyInterscience, New York · Zbl 0084.10402
[9] Kohsaka, F.; Takahashi, W., Proximal point algorithms with Bregman functions in Banach spaces, J. Nonlinear Convex Anal., 6, 505-523 (2005) · Zbl 1105.47059
[10] Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces II (1979), Springer: Springer, Berlin · Zbl 0403.46022 · doi:10.1007/978-3-662-35347-9
[11] Maingé, P. E., The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 59, 1, 74-79 (2010) · Zbl 1189.49011 · doi:10.1016/j.camwa.2009.09.003
[12] Maingé, P. E., Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16, 899-912 (2008) · Zbl 1156.90426 · doi:10.1007/s11228-008-0102-z
[13] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4
[14] Phelps, R. P., Convex Functions, Monotone Operators, and Differentiability, 1364 (1993), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0921.46039
[15] Qu, B.; Xiu, N., A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21, 5, 1655-1665 (2005) · Zbl 1080.65033 · doi:10.1088/0266-5611/21/5/009
[16] Schöpfer, F.
[17] Schöpfer, F.; Schuster, T.; Louis, A. K., An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems, 24 (2008) · Zbl 1153.46308 · doi:10.1088/0266-5611/24/5/055008
[18] Shehu, Y., Iterative Methods for Split Feasibility Problems in certain Banach Spaces, J. Nonlinear Convex Anal., 16, 12, 2351-2364 (2015) · Zbl 1334.49037
[19] Shehu, Y., Strong convergence theorem for Multiple Sets Split Feasibility Problems in Banach Spaces, Numerical Funct. Anal. Optim., 37, 8, 1021-1036 (2016) · Zbl 1351.49043 · doi:10.1080/01630563.2016.1185614
[20] Shehu, Y.; Iyiola, O. S.; Enyi, C. D., An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces, Numer. Algor., 72, 835-864 (2016) · Zbl 1346.49051
[21] Shehu, Y.; Mewomo, O. T.; Ogbuisi, F. U., Further investigation into approximation of a common solution of fixed point problems and split feasibility problems, Acta Math. Sci., 36, 3, 913-930 (2016) · Zbl 1363.47118 · doi:10.1016/S0252-9602(16)30049-2
[22] Shehu, Y.; Ogbuisi, F. U.; Iyiola, O. S., Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces, Optimization, 65, 2, 299-323 (2016) · Zbl 1347.49014 · doi:10.1080/02331934.2015.1039533
[23] Takahashi, W., Nonlinear Functional Analysis-Fixed Point Theory and Applications (2000), Yokohama Publishers Inc.: Yokohama Publishers Inc., Yokohama · Zbl 0997.47002
[24] Takahashi, W., Nonlinear functional analysis (2000), Yokohama Publishers: Yokohama Publishers, Yokohama · Zbl 0997.47002
[25] Wang, F., A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces, Numerical Functional Anal. Optim., 35, 99-110 (2014) · Zbl 1480.47102 · doi:10.1080/01630563.2013.809360
[26] Wang, F.; Xu, H. K., Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74, 4105-4111 (2011) · Zbl 1308.47079 · doi:10.1016/j.na.2011.03.044
[27] Wen, M.; Peng, J.; Tang, Y., A Cyclic and Simultaneous Iterative Method for Solving the Multiple-Sets Split Feasibility Problem, J. Optim. Theory Appl. · Zbl 1330.90081
[28] Xu, H.-K., Inequalities in Banach spaces with applications, Nonlinear Anal., 16, 2, 1127-1138 (1991) · Zbl 0757.46033 · doi:10.1016/0362-546X(91)90200-K
[29] Xu, H.-K., A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22, 6, 2021-2034 (2006) · Zbl 1126.47057 · doi:10.1088/0266-5611/22/6/007
[30] Yang, Q., The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20, 4, 1261-1266 (2004) · Zbl 1066.65047 · doi:10.1088/0266-5611/20/4/014
[31] Yang, Q.; Zhao, J., Generalized KM theorems and their applications, Inverse Problems, 22, 3, 833-844 (2006) · Zbl 1117.65081 · doi:10.1088/0266-5611/22/3/006
[32] Yang, Y.; Zhang, S.; Yang, Q., Modified alternating direction methods for the modified multiple-sets split feasibility problems, J. Optim. Theory Appl., 163, 1, 130-147 (2014) · Zbl 1317.90286 · doi:10.1007/s10957-013-0502-6
[33] Yao, Y.; Jigang, W.; Liou, Y.-C.
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